Introduction

Today we will discuss a
not-so-famous method of inverting matrices. This method is recursive in
the sense that given a method to find inverse of square matrix of order
$ n$ it can be applied to find the inverse of a matrix of order
$ (n + 1)$. This method is named Partition Method or the Escalator Method.
The idea is to partition a matrix into smaller sub-matrices and then
calculate the inverse from the given inverse of one of the smaller
sub-matrices.

After the widely read post Two Problems from IIT-JEE, I am going to discuss two problems which are not from IIT-JEE (as far as I am aware). They are taken from the masterpiece "A Course of Pure Mathematics"
by G. H. Hardy. The first one is a tough limit problem (at least I have
not been able to find a simpler solution till now) and the second one
is an instructive example which deals with the behavior of derivatives
for large values of the argument.

Oscillation of a Function

In a previous post we obtained the Riemann's condition of integrability using the concept of upper and lower Darboux sums. We observed that in order that a function be Riemann integrable on interval $ [a, b]$ it was
necessary (and sufficient) to make the sum
$$U(P, f) - L(P, f) = \sum_{k = 1}^{n}(M_{k} - m_{k})(x_{k} - x_{k - 1})$$
arbitrarily small for some partition $ P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\}$ of $ [a, b]$.

In the last post we defined the Riemann integral of a function on a closed interval and discussed some of the conditions for the integrability of a function. Here we develop the full machinery of the Riemann integral starting with the basic properties first.